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Singular Schrödinger operators as self-adjoint extensions of $ N$-entire operators

Authors: Luis O. Silva, Gerald Teschl and Julio H. Toloza
Journal: Proc. Amer. Math. Soc. 143 (2015), 2103-2115
MSC (2010): Primary 34L40, 47B25; Secondary 46E22, 34B20
Published electronically: December 18, 2014
MathSciNet review: 3314119
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Abstract: We investigate the connections between Weyl-Titchmarsh-
Kodaira theory for one-dimensional Schrödinger operators and the theory of $ n$-entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional Schrödinger operator to be $ n$-entire in terms of square integrability of derivatives (w.r.t. the spectral parameter) of the Weyl solution. We also show that this is equivalent to the Weyl function being in a generalized Herglotz-Nevanlinna class. As an application we show that perturbed Bessel operators are $ n$-entire, improving the previously known conditions on the perturbation.

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Luis O. Silva
Affiliation: Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, C.P. 04510, México D.F.

Gerald Teschl
Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria — and — International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

Julio H. Toloza
Affiliation: CONICET — and — Centro de Investigación en Informática para la Ingeniería, Universidad Tecnológica Nacional – Facultad Regional Córdoba, Maestro López s/n, X5016ZAA Córdoba, Argentina

Keywords: Schr\"odinger operators, de Branges spaces, Weyl--Titchmarsh--Kodaira theory
Received by editor(s): October 23, 2013
Published electronically: December 18, 2014
Additional Notes: The authors’ research was supported by the Austrian Science Fund (FWF) under Grant No. Y330 and by CONICET (Argentina) through grant PIP 112-201101-00245
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society